
A formula due to Mardia and Jupp [1]} is given in
[2]}(4.2)
as
\(\hat{\kappa }\approx d \bar{r} \left(1 + \frac{d}{d+2} \bar{r}^2 + \frac{d^2(d+8)}{(d+2)^2(d+4)}\bar{r}^4 \right)~.\) 
For large values of \(d\) , this is almost the same as
\(\hat{\kappa }\approx d \bar{r} \left(1 + \bar{r}^2 + \bar{r}^4 \right) ~.\) 

Banerjee [2]} derive an estimate (unnumbered, above [2]} (4.4)),
\(\hat{\kappa }\approx \frac{d \bar{r}}{1 - \bar{r}^2}\) 
which has series expansion
\(\hat{\kappa }\approx d \bar{r} (1 + \bar{r}^2 + \bar{r}^4 + \ldots ) ~.\) 
Since \(0 \le \bar{r} < 1\) , this series will converge, albeit slowly
for \(\bar{r}\)  close to 1.
Thus, it is seen that eq:order-2 is simply a truncated approximation
to the infinite series eq:order-infinity.
We shall refer to eq:order-2 (perhaps somewhat inexactly) as the
“order-2” approximation to eq:order-infinity, since the approximation
to \(1 / (1 - \bar{r}^2)\)  contains terms up to second order in \(\bar{r}^2\) .

It is also possible to consider approximations of other orders for
eq:order-frac, including in particular the 0-order approximation
\(\hat{\kappa }\approx d \bar{r} ~,\) 
first order approximation
\(\hat{\kappa }\approx d \bar{r} \left(1 + \bar{r}^2 \right) ~,\) 
and the third-order approximation.

Another empirically derived formula is also given in
[2]}(4.4). However, we observe (see fig:orders)
that the approximation
eq:order-frac is already a very close approximation, and the use of
[2]}(4.4) is not warranted.

